Finite Group Databasebeta

← #17086206

Subgroups — #17086206

Order 4352.

774 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763110C211yesmaximal, characteristic
2#14534172C211yesmaximal, characteristic, complemented
3#14534142C211yesmaximal, characteristic
4#14530282C211yesmaximal, characteristic, complemented
5#14530277C211yesmaximal, characteristic, complemented
6#14530171C211yesmaximal, characteristic
7#14530157C211yesmaximal, characteristic
8#14529749C211yesmaximal, characteristic, complemented
9#14529554C211yesmaximal, characteristic, complemented
10#14529517C211yesmaximal, characteristic, complemented
11#14529512C211yesmaximal, characteristic, complemented
12#14529467C211yesmaximal, characteristic
13#14529467C211yesmaximal, characteristic
14C4×#10587492C211yesmaximal, characteristic
15C2×C8×Q8×C17C211yesmaximal, characteristic, complemented
16C2×#10587556C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18#11763110—24no—
19C2×#10587489C2211yescharacteristic
20C2×#10587489C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22C2×#10587488C2211yescharacteristic
23C2×#10587492C2211yescharacteristic
24C2×#10587492C2211yescharacteristic
25C2×#10587492C2211yescharacteristic
26C2×#10587492C2211yescharacteristic
27#11762744—21no—
28#11762740—21no—
29#11762728—21no—
30#11762720—21no—
31#11762711C2211yescharacteristic
32#11762711C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762704—21no—
40#11762704—21no—
41#11762680—21no—
42#11762680—21no—
43#11762675C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53C2×C4×Q8×C17C2211yescharacteristic
54C8×Q8×C17—24no—
55C2×C17×C4⋊C8C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×C4⋊C8C2211yescharacteristic
58C2×C4×C8×C17C2211yescharacteristic
59C2×C4×C8×C17C2211yescharacteristic
60C2×C4×C8×C17C2211yescharacteristic
61C22×Q8×C17D411yescharacteristic
62C22×D68C2311yescharacteristic
63C22×C4×D17—21no—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C4×Q8×C17—22no—
69C4×Q8×C17—22no—
70C2×C4⋊C4×C17D411yescharacteristic
71C2×C4⋊C4×C17D411yescharacteristic
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×#63227—21no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×#63227—21no—
80C2×#63227—21no—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C4×D68—22no—
98C4×D68—22no—
99#10587556—22no—
100#10587556—22no—
101C17×C4⋊C8—22no—
102C17×C4⋊C8—22no—
103C17×C4⋊C8D414yes—
104C4×C8×C17—22no—
105C4×C8×C17—22no—
106C4×C8×C17—22no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587492—22no—
117#10587492—22no—
118#10587492—22no—
119#10587492Q814yescomplemented
120#10587489—22no—
121#10587489—22no—
122#10587488—22no—
123#10587488—22no—
124C23×D17—21no—
125C2×Q8×C17—22no—
126C2×Q8×C17—22no—
127C2×Q8×C17—22no—
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2411yescharacteristic
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68—21no—
146C2×D68—21no—
147C2×D68C2×Q812yes—
148C2×C4×D17—42no—
149C2×C4×D17—81no—
150C2×C4×D17—81no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17C2×D412yes—
161C2×C8×C17C2×D412yes—
162C2×C8×C17C2×Q812yes—
163C4⋊C4×C17—22no—
164C4⋊C4×C17—22no—
165C4⋊C4×C17—22no—
166C42×C17—21no—
167C42×C17—21no—
168C42×C17—21no—
169C42×C17—21no—
170C42×C17C2×D412yes—
171C42×C17#4112yes—
172#63227—42no—
173#63227—42no—
174#63227—42no—
175#63227—81no—
176#63227—81no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17C2×Q812yes—
190C4⋊Dic17#4112yes—
191#63225—42no—
192#63225—42no—
193#63225—42no—
194#63225—42no—
195#63225—81no—
196#63225—81no—
197#63225—81no—
198#63225—81no—
199C4×Dic17—81no—
200#25473—171nomaximal, Sylow
201C23×C17C22×D411yescharacteristic
202C22×D17—22no—
203C22×D17—41no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208C22×D17—41no—
209Q8×C17—28no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17#12112yes—
225C2×C4×C17#12112yes—
226C2×C4×C17#13512yes—
227C2×C4×C17#13512yes—
228C2×C4×C17C22×D411yescharacteristic
229C2×C4×C17C22×Q811yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C172−1+411yescharacteristic
233C2×C4×C172−1+411yescharacteristic
234C2×Dic17—22no—
235C2×Dic17—22no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255D68—22no—
256D68—22no—
257D68—41no—
258D68—41no—
259C4×D17—84no—
260C8×C17—22no—
261C8×C17—22no—
262C8×C17—22no—
263C8×C17—41no—
264C2×#478—171no—
265C2×C8×Q8D1711yescharacteristic, complemented
266#2584—171no—
267#2580—171no—
268#2088—171no—
269#2087—171no—
270#2067—171no—
271#2065—171no—
272#1991—171no—
273#1949—171no—
274#1942—171no—
275#1941—171no—
276#1934—171no—
277#1934—171no—
278C4×#102—171no—
279C22×C17C2×#12111yescharacteristic
280C22×C17#54811yescharacteristic
281C22×C17D4×Q811yescharacteristic
282C22×C17C2×#13511yescharacteristic
283C22×C17C2×#13511yescharacteristic
284C22×C17#57611yescharacteristic
285C22×C17#57711yescharacteristic
286C2×D17—42no—
287C2×D17—42no—
288C2×D17—42no—
289C2×D17—81no—
290C2×D17—81no—
291C2×D17—81no—
292C2×D17—81no—
293C4×C17—22no—
294C4×C17—22no—
295C4×C17—22no—
296C4×C17—22no—
297C4×C17—41no—
298C4×C17—41no—
299C4×C17#53311yescharacteristic
300C4×C17#54111yescharacteristic
301C4×C17#54711yescharacteristic
302C4×C17D4×Q811yescharacteristic
303Dic17—42no—
304Dic17—42no—
305Dic17—81no—
306Dic17—81no—
307Dic17—81no—
308Dic17—81no—
309Dic17—81no—
310C2×#125—171no—
311C2×C4×Q8C2×D1711yescharacteristic
312C2×C4×D4—171no—
313#478—344no—
314C8×Q8—24no—
315C2×#107—171no—
316C2×#107—171no—
317C2×#106—171no—
318C2×#106—171no—
319C2×C4⋊C8C2×D1711yescharacteristic
320C2×C4⋊C8C2×D1711yescharacteristic
321C2×C4⋊C8C2×D1711yescharacteristic
322C2×#102—171no—
323C2×#102—171no—
324C2×#102—171no—
325C2×#102—171no—
326C2×C4×C8C2×D1711yescharacteristic
327C2×C4×C8C2×D1711yescharacteristic
328C2×C4×C8C2×D1711yescharacteristic
329#399—341no—
330#398—341no—
331#395—341no—
332#393—341no—
333#390—171no—
334#390—171no—
335#389—171no—
336#389—171no—
337#389—171no—
338#389—171no—
339#389—171no—
340#389—171no—
341#389—341no—
342#389—341no—
343#385—341no—
344#385—341no—
345#382—171no—
346C4×C4⋊C4—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355C2×C17#242911yescharacteristic
356C2×C17#305011yescharacteristic
357C2×C17#306211yescharacteristic
358C2×C17#336111yescharacteristic
359C2×C17#336511yescharacteristic
360C2×C17#338011yescharacteristic
361C2×C17#338111yescharacteristic
362D17—82no—
363C22×Q8D6811yescharacteristic
364C22×D4—171no—
365C23×C4—341no—
366C22×C8C22×D1711yescharacteristic
367C22×C8C22×D1711yescharacteristic
368C22×C8C22×D1711yescharacteristic
369C22×C8C22×D1711yescharacteristic
370#125—342no—
371#125—342no—
372C4×Q8—22no—
373C4×Q8—22no—
374C4×D4—342no—
375C4×D4—342no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4D6811yescharacteristic
392C2×C4⋊C4D6811yescharacteristic
393C2×C4⋊C4D6811yescharacteristic
394C2×C22⋊C4—341no—
395C2×C22⋊C4—341no—
396C2×C22⋊C4—341no—
397C2×C22⋊C4—341no—
398C2×C22⋊C4—341no—
399C2×C42—341no—
400C2×C42C22×D1711yescharacteristic
401C2×C42C22×D1711yescharacteristic
402C2×C42C22×D1711yescharacteristic
403#107—342no—
404#107—342no—
405#106—342no—
406#106—342no—
407C4⋊C8—22no—
408C4⋊C8—22no—
409C4⋊C8C22⋊D1714yes—
410#102—174no—
411#102—342no—
412#102—342no—
413#102—342no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8—22no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426C17#2547311yescharacteristic, complemented, Sylow
427C24—341no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×D4—172no—
432C2×D4—341no—
433C2×D4—341no—
434C2×D4—341no—
435C2×D4—341no—
436C2×D4—341no—
437C22×C4—341no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4—341no—
441C22×C4—341no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4—682no—
445C22×C4—1361no—
446C22×C4—1361no—
447C22×C4C2×D6811yescharacteristic
448C22×C4C2×D6811yescharacteristic
449C22×C4C2×D6811yescharacteristic
450C22×C4C2×D6811yescharacteristic
451C22×C4C2×D6811yescharacteristic
452C22×C4C2×D6811yescharacteristic
453C22×C4C23×D1711yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8Q8×D1712yes—
464C2×C8C2×C22⋊D1712yes—
465C2×C8C2×C22⋊D1712yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—172no—
470C4⋊C4—172no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—682no—
484C4⋊C4—682no—
485C4⋊C4—682no—
486C4⋊C4—682no—
487C4⋊C4—1361no—
488C4⋊C4—1361no—
489C4⋊C4—1361no—
490C4⋊C4—1361no—
491C22⋊C4—682no—
492C22⋊C4—682no—
493C22⋊C4—682no—
494C22⋊C4—1361no—
495C22⋊C4—1361no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—1361no—
501C42#6325212yes—
502C42C2×C22⋊D1712yes—
503C23—342no—
504C23—681no—
505C23—681no—
506C23—681no—
507C23—681no—
508C23—681no—
509C23—681no—
510C23C22×D6811yescharacteristic
511Q8—28no—
512D4—342no—
513D4—342no—
514D4—681no—
515D4—681no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—342no—
531C2×C4—342no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—1364no—
552C2×C4#1058759012yes—
553C2×C4#1058759012yes—
554C2×C4#1058761612yes—
555C2×C4#1058761612yes—
556C2×C4C2×#6325211yescharacteristic
557C2×C4C2×Q8×D1711yescharacteristic
558C2×C4#1058769411yescharacteristic
559C2×C4#41×D1711yescharacteristic
560C2×C4#1058769711yescharacteristic
561C2×C4C22×C22⋊D1711yescharacteristic
562C8—22no—
563C8—22no—
564C8—22no—
565C8—41no—
566C22—682no—
567C22—682no—
568C22—682no—
569C22—1361no—
570C22—1361no—
571C22—1361no—
572C22—1361no—
573C22C2×#1058761611yescharacteristic
574C22#1176347111yescharacteristic
575C22Q8×D6811yescharacteristic
576C22C2×#1058759011yescharacteristic
577C22C2×#1058759011yescharacteristic
578C22#1176361411yescharacteristic
579C22#1176362111yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—682no—
587C4—682no—
588C4—1361no—
589C4—1361no—
590C4—1361no—
591C4—1361no—
592C4—1361no—
593C4#1176336311yescharacteristic
594C4#1176341711yescharacteristic
595C4#1176345811yescharacteristic
596C4Q8×C22⋊D1711yescharacteristic
597C2—1362no—
598C2#1453237711yescharacteristic
599C2#1453836511yescharacteristic
600C2#1453846411yescharacteristic
601C2#1454144611yescharacteristic
602C2#1454148211yescharacteristic
603C2#1454165211yescharacteristic
604C2#1454166111yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C8×Q8×C17C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenterC2×C8×Q8D17
hypocenterC17#25473
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14534172C2semidirect
4. #14530282C2semidirect
5. #14530277C2semidirect
8. #14529749C2semidirect
9. #14529554C2semidirect
10. #14529517C2semidirect
11. #14529512C2semidirect
15. C2×C8×Q8×C17C2semidirect
119. #10587492Q8semidirect
265. C2×C8×Q8D17semidirect
426. C17#25473semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#117631102
#145341722
#145341422
#145302822
#145302772
#145301712
#145301572
#145297492
#145295542
#145295172
#145295122
#145294672
C4×#105874922
C2×C8×Q8×C172
#2547317